Problem: A composite function can be written as $w\bigl(u(x)\bigr)$, where $u$ and $w$ are basic functions. Is $g(x)=\sqrt{e^x}$ a composite function? If so, what are the "inner" and "outer" functions? Choose 1 answer: Choose 1 answer: (Choice A) A $g$ is composite. The "inner" function is $e^x$ and the "outer" function is $\sqrt{x}$. (Choice B) B $g$ is composite. The "inner" function is $\sqrt{x}$ and the "outer" function is $e^x$. (Choice C) C $g$ is not a composite function.
Composite and combined functions A composite function is where we make the output from one function, in this case $u$, the input for another function, in this case $w$. We can also combine functions using arithmetic operations, but such a combination is not considered a composite function. The inner function The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function. Here, we have $e^x$ inside the radical. We evaluate this expression first, so $u(x)=e^x$ is the inner function. The outer function Then we take the square root of the entire output of $u$. So $w(x)=\sqrt{x}$ is the outer function. Answer $g$ is composite. The "inner" function is $e^x$ and the "outer" function is $\sqrt{x}$. Note that there are other valid ways to decompose $g$, especially into more complicated functions.